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\title{《基础复分析》第6章留数计算 - 习题}
\author{CGZ ET AL}

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%## 《基础复分析》习题六

\begin{enumerate}

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\item % 1

求下列函数的极点与留数:
- $\dfrac{1}{z^2 + 5z + 6}$
- $\dfrac{1}{(z^2 - 1)^2}$
- $\dfrac{1}{\sin z}$
- $\cot z$
- $\dfrac{1}{\sin^2 z}$
- $\dfrac{1}{z^m (1-z)^n}$ ($n, m$ 为正整数)
    

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\item % 2

设 $P(z)$ 为 $n \geq 2$ 次多项式, 证明 $1/P(z)$ 在所有有限极点的留数之和为零.
    

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\item % 3

设 $P(z)$ 为 $n \geq 2$ 次多项式, $f(z)$ 为有理函数. 证明 $f(P(z))$ 在所有有限极点的留数之和为零.
    

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\item % 4

设 $P(z)$ 是一个首项系数为 $1$ 的 $n \geq 2$ 次多项式, 设区域 $\Omega$ 的一个余集分支包含 $P(z)$ 的所有零点. 证明 $\sqrt[n]{P(z)}$ 在 $\Omega$ 内可以定义解析单值分支. 并对 $\Omega$ 内的任意一条闭曲线 $\gamma$, 求积分
    $$
    \int_{\gamma} \frac{dz}{\sqrt[n]{P(z)}}
    $$
    的可能的值.
    

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\item % 5

求多项式 $z^7 - 2z^5 + 6z^3 - z + 1$ 在单位圆盘内的零点个数.
    

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\item % 6

求多项式 $z^4 - 6z + 3$ 在圆环 $1 < |z| < 2$ 内的零点个数.
    

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\item % 7

求多项式 $z^4 + 8z^3 + 3z^2 + 8z + 3$ 在右半平面的零点个数.
    

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\item % 8

用留数定理计算下列积分:

- $\displaystyle \int_0^{\frac{\pi}{2}} \frac{dx}{a + \sin^2 x}$ ($|a| > 1$)

- $\displaystyle \int_0^{+\infty} \frac{x^2 \, dx}{x^4 + 5x^2 + 6}$

- $\displaystyle \int_{-\infty}^{+\infty} \frac{x^2 - x + 2}{x^4 + 10x^2 + 9} \, dx$

- $\displaystyle \int_0^{+\infty} \frac{x^2 \, dx}{(x^2 + a^2)^2}$ ($a$ 为实数)

- $\displaystyle \int_0^{+\infty} \frac{\cos x}{x^2 + a^2} \, dx$ ($a$ 为实数)

- $\displaystyle \int_0^{+\infty} \frac{x \sin x}{x^2 + a^2} \, dx$ ($a$ 为实数)

- $\displaystyle \int_0^{+\infty} \frac{x^{\frac{1}{3}}}{1 + x^2} \, dx$

- $\displaystyle \int_0^{+\infty} \frac{\ln^2 x}{(1 + x^2)^2} \, dx$
    

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\item % 9

用留数定理与分部积分法计算积分.
    $$
    \int_0^{+\infty} \frac{\ln(1+x^2)}{x^{1+\alpha}} \, dx \quad (0 < \alpha < 2)
    $$
    

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\item % 10

(Bergman (伯格曼) 公式) 设 $f(z)$ 是单位圆盘内的有界解析函数. 证明对 $|z| < 1$,
    $$
    f(\zeta) = \frac{1}{\pi} \iint_{|z|<1} \frac{f(z) \, dx \, dy}{(1 - z\overline{\zeta})^2}.
    $$

(提示: 将面积分表示为极坐标, 然后把内层积分变成线积分.)

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\end{enumerate}

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